Study On Three-dimensional Direct Current Resistivity Forward Modeling Based On Balanced Domain Decomposition Method With Constraints

In general,the three-dimensional numerical simulation of direct current(DC)resistivity is discretized by numerical method(like finite difference)to obtain linear equations and solve them.However,the memory and computation time required by the current mainstream solutions of linear equations increase exponentially with the increase of mesh number,which results in more computing resources for the DC resistivity three-dimensional fine inversion of large data volume,and restricting the improvement of computing speed.In recent years,domain decomposition method,as a powerful means to solve large-scale complex numerical problems,has been applied to geophysical electromagnetic methods.However,there are many kinds of domain decomposition methods,and the factors affecting the computational efficiency of the method are complex.So,balanced domain decomposition method with constraints(BDDC)was introduced into the three-dimensional finite difference numerical simulation of DC resistivity in this paper,the computational efficiency and influencing factors of the BDDC method were mainly studied,which can provide reference for largescale fast forward and inversion of electromagnetic methods.Starting from the basic equation satisfied by DC resistivity method,the large sparse linear equations were obtained by using finite difference discretization combined with boundary conditions,the computed region was decomposed into several subdomains by BDDC method,and the subdomain problems and boundary equation were solved by iterative method,accelerated the solving speed of three-dimensional forward electromagnetic method.Then,the horizontal two-layer model and the vertical contact zone model were designed to verified the accuracy and feasibility of the method,discussed and analyzed the influence of the number of subdomains,the method of solving subdomain problems and boundary equation,and the mesh number on the computational efficiency of BDDC method.Finally,BDDC method was used to simulate leak monitoring of landfill model,a leakage monitoring device for launching in well and receiving in impermeable layer was proposed,and the factors affecting the leak monitoring were discussed.By comparing the results of several different theoretical models,the computation time of BDDC method decreases first and then increases with the increase of the number of subdomains,and the computation time is the least when the number of subdomains is close to the number of elements in a single subdomain.Among three kinds of iteration methods for subdomain problems and boundary equation adopted in this paper,BDDC method combined with PCG has the fastest computing speed,BICG takes the second place,and Steepest Descent takes the slowest,and the computation time of BDDC increases exponentially with the increase of mesh number.To sum up,BDDC method can improve the efficiency of three-dimensional forward modeling of DC resistivity,but the appropriate number of subdomains and the solution of subdomain problems and boundary equation should be selected.The accuracy of the detection device proposed in this paper is comparable to that of the electrical detection technology,and the spatial distribution of leakage can be roughly inferred.